The two clubs agreed a fee, understood to be £34million ($47m) plus add ons, on Monday and the deal is now subject to medical.
There was progress made in negotiations between the two clubs over the weekend, with agreements reached on personal terms and a final agreement reached on a fee on Monday.
Goal understands the base fee is £34m plus £7m ($9m) in add-ons, meaning United could shell out as much as £41m ($56m).
Real Madrid, meanwhile, will pay a solidarity payment of £1m to the defender’s former club Lens.
United have been working on a deal for the France international since concluding the transfer of Sancho. He has won the Champions League four times and the World Cup once and his winning mentality will be a boost to Solskjaer’s side.
His current deal with Madrid was due to expire next year but he made it clear to the Liga giants he did not want to sign fresh terms as he wanted a fresh challenge. Madrid knew if they did not sell this summer they risked losing him on a free transfer next year, as they did with Sergio Ramos this summer.